A note on Baer and Quasi-Baer modules

Abstract

We give new characterizations of Baer and quasi-Baer modules. We study the direct sums of (quasi-)Baer modules and we give a sufficient condition under which the direct sums of (quasi-)Baer modules has the (quasi-)Baer property. Also we extended some results from the rings theory to the modules theory. We study the endomorphism ring of Baer and quasi-Baer modules and we show that : If M is a quasi-injective R-module . Then M is Baer iff S = EndR(M) is Von Neumann regular ring. Also we show that if M is an R-module whose endomorphism ring S = EndR(M) is extending, then M is Baer iff S is right nonsingular ring. As another we show that if M is an R-module such that S = EndR(M) has no infinite set of non-zero orthogonal idempotents. Then M is Baer iff S is right Rickart ring